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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.lognormal_dist"></a><a class="link" href="lognormal_dist.html" title="Log Normal Distribution">Log Normal
        Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">lognormal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">lognormal_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">lognormal_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">lognormal</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">lognormal_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
   <span class="comment">// Construct:</span>
   <span class="identifier">lognormal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
   <span class="comment">// Accessors:</span>
   <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
   <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          The lognormal distribution is the distribution that arises when the logarithm
          of the random variable is normally distributed. A lognormal distribution
          results when the variable is the product of a large number of independent,
          identically-distributed variables.
        </p>
<p>
          For location and scale parameters <span class="emphasis"><em>m</em></span> and <span class="emphasis"><em>s</em></span>
          it is defined by the probability density function:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/lognormal_ref.svg"></span>

          </p></blockquote></div>
<p>
          The location and scale parameters are equivalent to the mean and standard
          deviation of the logarithm of the random variable.
        </p>
<p>
          The following graph illustrates the effect of the location parameter on
          the PDF, note that the range of the random variable remains [0,+∞] irrespective
          of the value of the location parameter:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf1.svg" align="middle"></span>

          </p></blockquote></div>
<p>
          The next graph illustrates the effect of the scale parameter on the PDF:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf2.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.lognormal_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.member_functions"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">lognormal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs a lognormal distribution with location <span class="emphasis"><em>location</em></span>
          and scale <span class="emphasis"><em>scale</em></span>.
        </p>
<p>
          The location parameter is the same as the mean of the logarithm of the
          random variate.
        </p>
<p>
          The scale parameter is the same as the standard deviation of the logarithm
          of the random variate.
        </p>
<p>
          Requires that the scale parameter is greater than zero, otherwise calls
          <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.lognormal_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.non_member_accessors"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variable is [0,+∞].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.lognormal_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.accuracy"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The lognormal distribution is implemented in terms of the standard library
          log and exp functions, plus the <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">error
          function</a>, and as such should have very low error rates.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.lognormal_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.implementation"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table <span class="emphasis"><em>m</em></span> is the location parameter
          of the distribution, <span class="emphasis"><em>s</em></span> is its scale parameter, <span class="emphasis"><em>x</em></span>
          is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
          = 1-p</em></span>.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = e<sup>-(ln(x) - m)<sup>2 </sup> / 2s<sup>2 </sup> </sup> / (x * s * sqrt(2pi))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = cdf(normal_distribtion&lt;RealType&gt;(m,
                    s), log(x))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = cdf(complement(normal_distribtion&lt;RealType&gt;(m,
                    s), log(x)))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = exp(quantile(normal_distribtion&lt;RealType&gt;(m,
                    s), p))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = exp(quantile(complement(normal_distribtion&lt;RealType&gt;(m,
                    s), q)))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    e<sup>m + s<sup>2 </sup> / 2 </sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    (e<sup>s<sup>2</sup> </sup> - 1) * e<sup>2m + s<sup>2 </sup> </sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    e<sup>m - s<sup>2 </sup> </sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    sqrt(e<sup>s<sup>2</sup> </sup> - 1) * (2 + e<sup>s<sup>2</sup> </sup>)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    e<sup>4s<sup>2</sup> </sup> + 2e<sup>3s<sup>2</sup> </sup> + 3e<sup>2s<sup>2</sup> </sup> - 3
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    e<sup>4s<sup>2</sup> </sup> + 2e<sup>3s<sup>2</sup> </sup> + 3e<sup>2s<sup>2</sup> </sup> - 6
                  </p>
                </td>
</tr>
</tbody>
</table></div>
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      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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